Find one value of $x$ that is a solution to the equation: $(x^2+4)^2-11(x^2+4)+24=0$ $x=$
We could solve for $x$ by expanding $(x^2+4)^2$ and $-11(x^2+4)$, combining terms that are alike, and using the quadratic formula or factoring to solve for $x$. However there is a more elegant way to approach this problem. Let's use structural features to rewrite the equation in a simpler form. Note that if we let ${p}={x^2+4}$, we can rewrite the equation: $({x^2+4})^2-11({x^2+4})+24=0$ In particular, we can express it in the form: ${p}^2-11{p}+24=0$ Let's solve this equation in terms of ${p}$ : $\begin{aligned}{p}^2-11{p}+24&=0\\\\ ({p}-8)({p}-3)&=0\\\\ {p}=8\ &\text{or} \ \ {p}=3 \end{aligned}$ Since ${p}={x^2+4}$, let's substitute this value back into our two solutions in order to solve for $x$ : ${x^2+4}=8\ \ \ \text{or} \ \ \ {x^2+4}=3$ When we solve ${x^2+4}=8$, we find that $x=\pm2$. Note that there are no real solutions to the equation ${x^2+4}=3$. [Why not?] In conclusion, the two solutions of the equation $(x^2+4)^2-11(x^2+4)+24=0$ are $x=2$ and $x=-2$ : [Is there another way to solve for x?]